On plus-one generated arrangements of plane conics

TL;DR

This paper studies special conic arrangements in the complex projective plane, introducing new tools to characterize plus-one generated arrangements with certain singularities, and classifies those with only nodes and tacnodes, also presenting new examples of arrangements with identical combinatorics but different algebraic properties.

Contribution

It introduces a novel characterization tool for plus-one generated conic arrangements and classifies arrangements with specific singularities, expanding understanding of their algebraic and combinatorial properties.

Findings

Characterization of plus-one generated conic arrangements

Classification of arrangements with nodes and tacnodes

Examples of strong Ziegler pairs with identical combinatorics

Abstract

In this paper, we examine the combinatorial properties of conic arrangements in the complex projective plane that possess certain quasi-homogeneous singularities. First, we introduce a new tool that enables us to characterize the property of being plus-one generated within the class of conic arrangements with some naturally chosen quasi-homogeneous singularities. Next, we present a classification result on plus-one generated conic arrangements admitting only nodes and tacnodes as singularities. Building on results regarding conic arrangements with nodes and tacnodes, we present new examples of strong Ziegler pairs of conic-line arrangements -- that is, arrangements having the same strong combinatorics but distinct derivation modules.